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A two-species weak competition system of reaction-diffusion-advection with double free boundaries
Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation
1. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
2. | Department of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China |
3. | College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China |
$\begin{eqnarray*}\left\{\begin{array}{llll}n_t+u·\nabla n = \nabla·(D(n)\nabla n)-\nabla·(n \mathcal{S}(x, n, c)·\nabla c)\\ +ξ n-μ n^{2}, &x∈ Ω, &t>0, \\c_{t}+u·\nabla c = Δ c-c+n, &x∈Ω, &t>0, \\u_{t}+\nabla P = Δ u+n\nablaφ, &x∈Ω, &t>0, \\\nabla· u = 0, &x∈Ω, &t>0\end{array}\right.\end{eqnarray*}$ |
$ \begin{equation*}{\label{1.3}}\begin{split}|\mathcal{S}(x, n, c)|\leq C_{\mathcal{S}}(1+n)^{-α}\end{split}\end{equation*}$ |
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show all references
References:
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K. Baghaei and M. Hesaaraki,
Global existence and boundedness of classical solutions for a chemotaxis model with logstic source, C. R. Acad. Sci. Paris. Ser. I, 351 (2013), 585-591.
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[2] |
V. Calvez and J. A. Carrillo,
Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175.
doi: 10.1016/j.matpur.2006.04.002. |
[3] |
X. Cao and S. Ishida,
Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913.
doi: 10.1088/0951-7715/27/8/1899. |
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T. Cieślak,
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T. Cieślak and C. Stinner,
Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[6] |
T. Cieślak and C. Stinner,
New critical exponents in a fully parabolic quasilinear system Keller-Segel system and applications to volume filling models, J. Differnetial Equations, 258 (2015), 2080-2113.
doi: 10.1016/j.jde.2014.12.004. |
[7] |
T. Cieślak and M. Winkler,
Finite-time blow-up in a quasilinear system of chemoatxis, Nonlinearity, 21 (2008), 1057-1076.
doi: 10.1088/0951-7715/21/5/009. |
[8] |
K. Djie and M. Winkler,
Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[9] |
K. Fujie,
Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.
doi: 10.1016/j.jmaa.2014.11.045. |
[10] |
M. Herrero and J. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.
|
[11] |
T. Hillen and K. J. Painter,
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doi: 10.1007/s00285-008-0201-3. |
[12] |
T. Hillen and K. J. Painter,
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doi: 10.1006/aama.2001.0721. |
[13] |
D. Horstmann and G. Wang,
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doi: 10.1017/S0956792501004363. |
[14] |
D. Horstmann and M. Winkler,
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doi: 10.1016/j.jde.2004.10.022. |
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D. Horstmann,
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|
[16] |
S. Ishida,
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doi: 10.3934/dcds.2015.35.3463. |
[17] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[18] |
S. Ishida and T. Yokota,
Blow-up finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Continuous Dynam. Systems - B, 18 (2013), 2569-2596.
doi: 10.3934/dcdsb.2013.18.2569. |
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Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
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[20] |
A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case, J. Math. Phys., 53 (2012), 115609, 9pp.
doi: 10.1063/1.4742858. |
[21] |
A. Kiselev and L. Ryzhik,
Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318.
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[22] |
R. Kowalczyk and Z. Szymańska,
On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.
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Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., 25 (2015), 721-746.
doi: 10.1142/S0218202515500177. |
[25] |
X. Li, Y. Wang and Z. Xiang,
Global existence and boundedness in a 2D KellerSegel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910.
doi: 10.4310/CMS.2016.v14.n7.a5. |
[26] |
X. Li and Y. Xiao,
Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Anal., 37 (2017), 14-30.
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[27] |
J. Liu and Y. Wang,
Global existence and boundedness in a Keller-Segel- (Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl., 447 (2017), 499-528.
doi: 10.1016/j.jmaa.2016.10.028. |
[28] |
J. Liu and Y. Wang,
Boundedness and decay property in a three-dimensionlal Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differential Equations, 261 (2016), 967-999.
doi: 10.1016/j.jde.2016.03.030. |
[29] |
T. Nagai,
Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
|
[30] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[31] |
K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
|
[32] |
Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux, Zeitschrift f/"ur angewandte Mathematik und Physik, 68 (2017), p68.
doi: 10.1007/s00033-017-0816-6. |
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Y. Sugiyama,
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|
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doi: 10.1007/s00033-016-0732-1. |
[35] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[36] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[37] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
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